Eisenman intrinsic measures and algebraic invariants

We generalize the Sakai theorem that says that every complex algebraic mani fold of general type is measure hyperbolic. We introduce the notion of k-me asure hyperbolicity for every Eisenman k-measure and, following Sakai, we c onsider an analogue <(kappa)over bar>k of the Kodaira logarithmic dimension which construction uses logarithmic k-forms. We show that a complex algebr aic manifold is k-measure hyperbolic if <(kappa)over bar>k(X) = dimX.